Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates

We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the Farey tesselation. Then we express the Hilbert transform and the Fourier coefficients of the Zygmund vector fields in terms of the above parametrization by infinitesimal shear functions. Finally, we compute the Weil-Petersson metric on the Teichm\"uller space of a punctured surface in terms of shears.